FOM: "beautiful"

[Harvey Friedman]

Since I am in the middle of major advances, it is not the best use of my
time at this particular moment for me to comment on the nontechnical
foundational nature of these recent postings. This will be done after
things have settled down.

However, I gather that several people are going to comment on this soon.

The word "beautiful" came about in connection with the reaction of some
extremely highly regarded and famous core mathematicians. In light of these
reactions and various reflective discussions concerning the use of the word
"beautiful" by mathematicians, it now appears that "beautiful" is likely to
attain a significant role in foundations of mathematics.

Although one cannot expect at this stage to give formal definitions of
"beautiful", it appears that there is a lot of objectivity to it, and it
can at least be approximated by attributes such as

succinct
clear
uncluttered
no ad hoc hypotheses or conclusions
symmetric
etcetera

Thus "beautiful" is much closer to a technical term than most people would
think.

"Beautiful" is considerably stronger than "natural", but it is in the same
general category as "natural".

So such questions as

"is there a natural statement in normal mathematics that is independent of ZFC"
"is there a natural statement in discrete or continuous mathematics that is
independent of ZFC"

can be sharpened to

"is there a beautiful statement in normal mathematics that is independent
of ZFC"
"is there a beautiful statement in discrete or continuous mathematics that
is independent of ZFC"

where these attributes "natural" and "beautiful" are subject to being
judged by the reactions of mathematicians of various levels of reputation.